package ‘financialmath’ - r · december 16, 2016 type package title financial mathematics for...

Package ‘FinancialMath’December 16, 2016

Type Package

Title Financial Mathematics for Actuaries

Version 0.1.1

Author Kameron Penn [aut, cre],Jack Schmidt [aut]

Maintainer Kameron Penn <kameron.penn.financialmath@gmail.com>

Description Contains financial math functions and introductory derivative functions in-cluded in the Society of Actuaries and Casualty Actuarial Society 'Financial Mathemat-ics' exam, and some topics in the 'Models for Financial Economics' exam.

License GPL-2

Encoding UTF-8

LazyData true

NeedsCompilation no

Repository CRAN

Date/Publication 2016-12-16 22:51:34

R topics documented:amort.period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2amort.table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4annuity.arith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5annuity.geo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7annuity.level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8bear.call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9bear.call.bls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11bls.order1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13bull.call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15bull.call.bls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16butterfly.spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17butterfly.spread.bls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18cf.analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 amort.period

collar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20collar.bls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22covered.call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23covered.put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25forward.prepaid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26IRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28NPV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29option.call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30option.put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31perpetuity.arith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32perpetuity.geo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34perpetuity.level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35protective.put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36rate.conv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37straddle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38straddle.bls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39strangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41strangle.bls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42swap.commodity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43swap.rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44TVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45yield.dollar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46yield.time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Index 49

amort.period Amortization Period

Description

Solves for either the number of payments, the payment amount, or the amount of a loan. Thepayment amount, interest paid, principal paid, and balance of the loan are given for a specifiedperiod.

amort.period(Loan=NA,n=NA,pmt=NA,i,ic=1,pf=1,t=1)

Arguments

Loan loan amount

n the number of payments/periods

pmt value of level payments

i nominal interest rate convertible ic times per year

ic interest conversion frequency per year

amort.period 3

pf the payment frequency- number of payments per year

t the specified period for which the payment amount, interest paid, principal paid,and loan balance are solved for

Details

Effective Rate of Interest: eff.i = (1 + iic )

ic − 1

j = (1 + eff.i)1pf − 1

Loan = pmt ∗ a n|jBalance at the end of period t: Bt = pmt ∗ a n−t|jInterest paid at the end of period t: it = Bt−1 ∗ jPrincipal paid at the end of period t: pt = pmt− it

Returns a matrix of input variables, calculated unknown variables, and amortization figures for thegiven period.

Assumes that payments are made at the end of each period.

One of n, pmt, or Loan must be NA (unknown).

If pmt is less than the amount of interest accumulated in the first period, then the function will stopbecause the loan will never be paid off due to the payments being too small.

If the pmt is greater than the loan amount plus interest accumulated in the first period, then thefunction will stop because one payment will pay off the loan.

t cannot be greater than n.

Author(s)

Kameron Penn and Jack Schmidt

See Also

amort.table

Examples

amort.period(Loan=100,n=5,i=.01,t=3)

amort.period(n=5,pmt=30,i=.01,t=3,pf=12)

amort.period(Loan=100,pmt=24,ic=1,i=.01,t=3)

4 amort.table

amort.table Amortization Table

Description

Produces an amortization table for paying off a loan while also solving for either the number ofpayments, loan amount, or the payment amount. In the amortization table the payment amount,interest paid, principal paid, and balance of the loan are given for each period. If n ends up notbeing a whole number, outputs for the balloon payment, drop payment and last regular payment areprovided. The total interest paid, and total amount paid is also given. It can also plot the percentageof each payment toward interest vs. period.

amort.table(Loan=NA,n=NA,pmt=NA,i,ic=1,pf=1,plot=FALSE)

Arguments

Loan loan amount

n the number of payments/periods

plot tells whether or not to plot the percentage of each payment toward interest vs.period

Details

ic − 1

j = (1 + eff.i)1pf − 1

Loan = pmt ∗ a n|jBalance at the end of period t: Bt = pmt ∗ a n−t|jInterest paid at the end of period t: it = Bt−1 ∗ jPrincipal paid at the end of period t: pt = pmt− it

Total Paid= pmt ∗ nTotal Interest Paid= pmt ∗ n− Loan

If n = n∗ + k where n∗ is an integer and 0 < k < 1:

Last regular payment (at period n∗) = pmt ∗ s k|jDrop payment (at period n∗ + 1) = Loan ∗ (1 + j)n

∗+1 − pmt ∗ s n∗|jBalloon payment (at period n∗) = Loan ∗ (1 + j)n

∗ − pmt ∗ s n∗|j + pmt

annuity.arith 5

A list of two components.

Schedule A data frame of the amortization schedule.

Other A matrix of the input variables and other calculated variables.

Assumes that payments are made at the end of each period.

One of n, Loan, or pmt must be NA (unknown).

If pmt is less than the amount of interest accumulated in the first period, then the function will stopbecause the loan will never be paid off due to the payments being too small.

If pmt is greater than the loan amount plus interest accumulated in the first period, then the functionwill stop because one payment will pay off the loan.

Author(s)

See Also

amort.period

annuity.level

Examples

amort.table(Loan=1000,n=2,i=.005,ic=1,pf=1)

amort.table(Loan=100,pmt=40,i=.02,ic=2,pf=2,plot=FALSE)

amort.table(Loan=NA,pmt=102.77,n=10,i=.005,plot=TRUE)

annuity.arith Arithmetic Annuity

Description

Solves for the present value, future value, number of payments/periods, amount of the first payment,the payment increment amount per period, and/or the interest rate for an arithmetically growingannuity. It can also plot a time diagram of the payments.

annuity.arith(pv=NA,fv=NA,n=NA,p=NA,q=NA,i=NA,ic=1,pf=1,imm=TRUE,plot=FALSE)

6 annuity.arith

Arguments

pv present value of the annuity

fv future value of the annuity

n number of payments/periods

p amount of the first payment

q payment increment amount per period

i nominal interest frequency convertible ic times per year

imm option for annuity immediate or annuity due, default is immediate (TRUE)

plot option to display a time diagram of the payments

Details

ic − 1

j = (1 + eff.i)1pf − 1

fv = pv ∗ (1 + j)n

Annuity Immediate:

pv = p ∗ a n|j + q ∗a n|j−n∗(1+j)

Annuity Due:

pv = (p ∗ a n|j + q ∗a n|j−n∗(1+j)

j ) ∗ (1 + i)

Returns a matrix of the input variables, and calculated unknown variables.

At least one of pv, fv, n, p, q, or i must be NA (unknown).

pv and fv cannot both be specified, at least one must be NA (unknown).

Author(s)

See Also

annuity.geo

annuity.level

perpetuity.arith

perpetuity.geo

perpetuity.level

annuity.geo 7

Examples

annuity.arith(pv=NA,fv=NA,n=20,p=100,q=4,i=.03,ic=1,pf=2,imm=TRUE)

annuity.arith(pv=NA,fv=3000,n=20,p=100,q=NA,i=.05,ic=3,pf=2,imm=FALSE)

annuity.geo Geometric Annuity

Description

Solves for the present value, future value, number of payments/periods, amount of the first payment,the payment growth rate, and/or the interest rate for a geometrically growing annuity. It can alsoplot a time diagram of the payments.

annuity.geo(pv=NA,fv=NA,n=NA,p=NA,k=NA,i=NA,ic=1,pf=1,imm=TRUE,plot=FALSE)

Arguments

fv future value of the annuity

n number of payments/periods for the annuity

k payment growth rate per period

pf the payment frequency- number of payments/periods per year

plot option to display a time diagram of the payments

Details

ic − 1

j = (1 + eff.i)1pf − 1

fv = pv ∗ (1 + j)n

Annuity Immediate:

j != k: pv = p ∗ 1−( 1+k1+j )

j = k: pv = p ∗ n1+j

Annuity Due:

j != k: pv = p ∗ 1−( 1+k1+j )

j−k ∗ (1 + j)

j = k: pv = p ∗ n

8 annuity.level

Returns a matrix of the input variables and calculated unknown variables.

At least one of pv, fv, n, pmt, k, or i must be NA (unknown).

See Also

annuity.arith

annuity.level

perpetuity.arith

perpetuity.geo

perpetuity.level

Examples

annuity.geo(pv=NA,fv=100,n=10,p=9,k=.02,i=NA,ic=2,pf=.5,plot=TRUE)

annuity.geo(pv=NA,fv=128,n=5,p=NA,k=.04,i=.03,pf=2)

annuity.level Level Annuity

Description

Solves for the present value, future value, number of payments/periods, interest rate, and/or theamount of the payments for a level annuity. It can also plot a time diagram of the payments.

annuity.level(pv=NA,fv=NA,n=NA,pmt=NA,i=NA,ic=1,pf=1,imm=TRUE,plot=FALSE)

Arguments

pv present value of the annuityfv future value of the annuityn number of payments/periodspmt value of the level paymentsi nominal interest rate convertible ic times per yearic interest conversion frequency per yearpf the payment frequency- number of payments/periods per yearimm option for annuity immediate or annuity due, default is immediate (TRUE)plot option to display a time diagram of the payments

bear.call 9

Details

ic − 1

j = (1 + eff.i)1pf − 1

Annuity Immediate:

pv = pmt ∗ a n|j = pmt ∗ 1−(1+j)−n

fv = pmt ∗ s n|j = pmt ∗ a n|j ∗ (1 + j)n

Annuity Due:

pv = pmt ∗ a n|j = pmt ∗ a n|j ∗ (1 + j)

fv = pmt ∗ s n|j = pmt ∗ a n|j ∗ (1 + j)n+1

At least one of pv, fv, n, pmt, or i must be NA (unknown).

See Also

annuity.arith

annuity.geo

perpetuity.arith

perpetuity.geo

perpetuity.level

Examples

annuity.level(pv=NA,fv=101.85,n=10,pmt=8,i=NA,ic=1,pf=1,imm=TRUE)

annuity.level(pv=80,fv=NA,n=15,pf=2,pmt=NA,i=.01,imm=FALSE)

bear.call Bear Call Spread

Description

Gives a table and graphical representation of the payoff and profit of a bear call spread for a rangeof future stock prices.

bear.call(S,K1,K2,r,t,price1,price2,plot=FALSE)

10 bear.call

Arguments

S spot price at time 0

K1 strike price of the short call

K2 strike price of the long call

r yearly continuously compounded risk free rate

t time of expiration (in years)

price1 price of the short call with strike price K1

price2 price of the long call with strike price K2

plot tells whether or not to plot the payoff and profit

Details

Stock price at time t = St

For St <= K1: payoff = 0

For K1 < St < K2: payoff = K1− St

For St >= K2: payoff = K1−K2

payoff = profit + (price1 - price2)∗er∗t

Payoff A data frame of different payoffs and profits for given stock prices.

Premiums A matrix of the premiums for the call options and the net cost.

K1 must be less than S, and K2 must be greater than S.

Author(s)

See Also

bear.call.bls

bull.call

option.call

Examples

bear.call(S=100,K1=70,K2=130,r=.03,t=1,price1=20,price2=10,plot=TRUE)

bear.call.bls 11

bear.call.bls Bear Call Spread - Black Scholes

Description

Gives a table and graphical representation of the payoff and profit of a bear call spread for a rangeof future stock prices. Uses the Black Scholes equation for the call prices.

bear.call.bls(S,K1,K2,r,t,sd,plot=FALSE)

Arguments

sd standard deviation of the stock (volatility)

Details

For K1 < St < K2: payoff = K1− St

payoff = profit+(priceK1 − priceK2) ∗ er∗t

Author(s)

12 bls.order1

See Also

bear.call

bull.call.bls

option.call

Examples

bear.call.bls(S=100,K1=70,K2=130,r=.03,t=1,sd=.2)

bls.order1 Black Scholes First-order Greeks

Description

Gives the price and first order greeks for call and put options in the Black Scholes equation.

bls.order1(S,K,r,t,sd,D=0)

Arguments

K strike price

r continuously compounded yearly risk free rate

D continuous dividend yield

A matrix of the calculated greeks and prices for call and put options.

Cannot have any inputs as vectors.

t cannot be negative.

Either both or neither of S and K must be negative.

Author(s)

bond 13

See Also

option.put

option.call

Examples

x <- bls.order1(S=100, K=110, r=.05, t=1, sd=.1, D=0)ThetaPut <- x["Theta","Put"]DeltaCall <- x[2,1]

bond Bond Analysis

Description

Solves for the price, premium/discount, and Durations and Convexities (in terms of periods). At aspecified period (t), it solves for the full and clean prices, and the write up/down amount. Also hasthe option to plot the convexity of the bond.

bond(f,r,c,n,i,ic=1,cf=1,t=NA,plot=FALSE)

Arguments

f face value

r coupon rate convertible cf times per year

c redemption value

n the number of coupons/periods for the bond

cf coupon frequency- number of coupons per year

t specified period for which the price and write up/down amount is solved for, ifnot NA

plot tells whether or not to plot the convexity

Details

ic − 1

j = (1 + eff.i)1cf − 1

coupon = f∗rcf (per period)

price = coupon∗a n|j + c ∗ (1 + j)−n

14 bond

MACD =

k=1k∗(1+j)−k∗coupon+n∗(1+j)−n∗c

MODD =

k=1k∗(1+j)−(k+1)∗coupon+n∗(1+j)−(n+1)∗c

MACC =

k=1k2∗(1+j)−k∗coupon+n2∗(1+j)−n∗c

MODC =

k=1k∗(k+1)∗(1+j)−(k+2)∗coupon+n∗(n+1)∗(1+j)−(n+2)∗c

Price (for period t):

If t is an integer: price =coupon∗a n−t|j + c ∗ (1 + j)−(n−t)

If t is not an integer then t = t∗ + k where t∗ is an integer and 0 < k < 1:

full price = ( coupon∗a n−t∗|j + c ∗ (1 + j)−(n−t∗)) ∗ (1 + j)k

clean price = full price−k∗coupon

If price > c :

premium = price−c

Write-down amount (for period t) = (coupon−c ∗ j) ∗ (1 + j)−(n−t+1)

If price < c :

discount = c−price

Write-up amount (for period t) = (c ∗ j−coupon) ∗ (1 + j)−(n−t+1)

A matrix of all of the bond details and calculated variables.

t must be less than n.

To make the duration in terms of years, divide it by cf.

To make the convexity in terms of years, divide it by cf2.

Examples

bond(f=100,r=.04,c=100,n=20,i=.04,ic=1,cf=1,t=1)

bond(f=100,r=.05,c=110,n=10,i=.06,ic=1,cf=2,t=5)

bull.call 15

bull.call Bull Call Spread

Description

Gives a table and graphical representation of the payoff and profit of a bull call spread for a rangeof future stock prices.

bull.call(S,K1,K2,r,t,price1,price2,plot=FALSE)

Arguments

Details

For K1 < St < K2: payoff = St −K1

profit = payoff + (price2 - price1)∗er∗t

16 bull.call.bls

See Also

bull.call.bls

bear.call

option.call

Examples

bull.call(S=115,K1=100,K2=145,r=.03,t=1,price1=20,price2=10,plot=TRUE)

bull.call.bls Bull Call Spread - Black Scholes

Description

Gives a table and graphical representation of the payoff and profit of a bull call spread for a rangeof future stock prices. Uses the Black Scholes equation for the call prices.

bull.call.bls(S,K1,K2,r,t,sd,plot=FALSE)

Arguments

Details

For K1 < St < K2: payoff = St −K1

profit = payoff+(priceK2 − priceK1) ∗ er∗t

butterfly.spread 17

See Also

bear.call

option.call

Examples

bull.call.bls(S=115,K1=100,K2=145,r=.03,t=1,sd=.2)

butterfly.spread Butterfly Spread

Description

Gives a table and graphical representation of the payoff and profit of a long butterfly spread for arange of future stock prices.

butterfly.spread(S,K1,K2=S,K3,r,t,price1,price2,price3,plot=FALSE)

Arguments

S spot price at time 0K1 strike price of the first long callK2 strike price of the two short callsK3 strike price of the second long callr continuously compounded yearly risk free ratet time of expiration (in years)price1 price of the long call with strike price K1price2 price of one of the short calls with strike price K2price3 price of the long call with strike price K3plot tells whether or not to plot the payoff and profit

Details

For K1 < St <= K2: payoff = St −K1

For K2 < St < K3: payoff = 2 ∗K2−K1− St

For St >= K3: payoff = 0

profit = payoff+(2∗price2 - price1 - price3) ∗ er∗t

18 butterfly.spread.bls

K2 must be equal to S.

K3 and K1 must both be equidistant to K2 and S.

K1 < K2 < K3 must be true.

See Also

butterfly.spread.bls

option.call

Examples

butterfly.spread(S=100,K1=75,K2=100,K3=125,r=.03,t=1,price1=25,price2=10,price3=5)

butterfly.spread.bls Butterfly Spread - Black Scholes

Description

Gives a table and graphical representation of the payoff and profit of a long butterfly spread for arange of future stock prices. Uses the Black Scholes equation for the call prices.

butterfly.spread.bls(S,K1,K2=S,K3,r,t,sd,plot=FALSE)

Arguments

K1 strike price of the first long call

K2 strike price of the two short calls

K3 strike price of the second long call

cf.analysis 19

Details

For K1 < St <= K2: payoff = St −K1

For K2 < St < K3: payoff = 2 ∗K2−K1− St

For St >= K3: payoff = 0

profit = payoff+(2 ∗ priceK2 − priceK1 − priceK3) ∗ er∗t

K2 must be equal to S.

K3 and K1 must both be equidistant to K2 and S.

K1 < K2 < K3 must be true.

See Also

butterfly.spread

option.call

Examples

butterfly.spread.bls(S=100,K1=75,K2=100,K3=125,r=.03,t=1,sd=.2)

cf.analysis Cash Flow Analysis

Description

Calculates the present value, macaulay duration and convexity, and modified duration and convexityfor given cash flows. It also plots the convexity and time diagram of the cash flows.

cf.analysis(cf,times,i,plot=FALSE,time.d=FALSE)

20 collar

Arguments

cf vector of cash flows

times vector of the periods for each cash flow

i interest rate per period

plot tells whether or not to plot the convexity

time.d tells whether or not to plot the time diagram of the cash flows

Details

pv =∑nk=1

cfk(1+i)timesk

MACD =

k=1timesk∗(1+i)−timesk∗cfk

MODD =

k=1timesk∗(1+i)−(timesk+1)∗cfk

MACC =

k=1timesk

2∗(1+i)−timesk∗cfkpv

MODC =

k=1timesk∗(timesk+1)∗(1+i)−(timesk+2)∗cfk

A matrix of all of the calculated values.

The periods in t must be positive integers.

See Also

Examples

cf.analysis(cf=c(1,1,101),times=c(1,2,3),i=.04,time.d=TRUE)

cf.analysis(cf=c(5,1,5,45,5),times=c(5,4,6,7,5),i=.06,plot=TRUE)

collar Collar Strategy

Description

Gives a table and graphical representation of the payoff and profit of a collar strategy for a range offuture stock prices.

collar 21

collar(S,K1,K2,r,t,price1,price2,plot=FALSE)

Arguments

K1 strike price of the long put

price1 price of the long put with strike price K1

Details

For St <= K1: payoff = K1− St

For K1 < St < K2: payoff = 0

For St >= K2: payoff = K2− St

profit = payoff + (price2 - price1)∗er∗t

Premiums A matrix of the premiums for the call and put options and the net cost.

See Also

collar.bls

option.put

option.call

Examples

collar(S=100,K1=90,K2=110,r=.05,t=1,price1=5,price2=15,plot=TRUE)

22 collar.bls

collar.bls Collar Strategy - Black Scholes

Description

Gives a table and graphical representation of the payoff and profit of a collar strategy for a range offuture stock prices. Uses the Black Scholes equation for the call and put prices.

collar.bls(S,K1,K2,r,t,sd,plot=FALSE)

Arguments

Details

For St >= K2: payoff = K2− St

profit = payoff+(priceK2 − priceK1) ∗ er∗t

Premiums A matrix of the premiums for the call and put options and the net cost.

See Also

option.put

option.call

Examples

collar.bls(S=100,K1=90,K2=110,r=.05,t=1,sd=.2)

covered.call 23

covered.call Covered Call

Description

Gives a table and graphical representation of the payoff and profit of a covered call strategy for arange of future stock prices.

covered.call(S,K,r,t,sd,price=NA,plot=FALSE)

Arguments

K strike price

price specified call price if the Black Scholes pricing is not desired (leave as NA touse the Black Scholes pricing)

Details

For St <= K: payoff = St

For St > K: payoff = K

profit = payoff + price∗er∗t − S

Premium The price of the call option.

Finds the put price by using the Black Scholes equation by default.

See Also

option.call

covered.put

24 covered.put

Examples

covered.call(S=100,K=110,r=.03,t=1,sd=.2,plot=TRUE)

covered.put Covered Put

Description

Gives a table and graphical representation of the payoff and profit of a covered put strategy for arange of future stock prices.

covered.put(S,K,r,t,sd,price=NA,plot=FALSE)

Arguments

K strike price

price specified put price if the Black Scholes pricing is not desired (leave as NA touse the Black Scholes pricing)

Details

For St <= K: payoff = S −K

For St > K: payoff = S − St

profit = payoff + price∗er∗t

Premium The price of the put option.

forward 25

See Also

option.put

covered.call

Examples

covered.put(S=100,K=110,r=.03,t=1,sd=.2,plot=TRUE)

forward Forward Contract

Description

Gives a table and graphical representation of the payoff of a forward contract, and calculates theforward price for the contract.

forward(S,t,r,position,div.structure="none",dividend=NA,df=1,D=NA,k=NA,plot=FALSE)

Arguments

position either buyer or seller of the contract ("long" or "short")

div.structure the structure of the dividends for the underlying ("none", "continuous", or "dis-crete")

dividend amount of each dividend, or amount of first dividend if k is not NA

df dividend frequency- number of dividends per year

k dividend growth rate per df

plot tells whether or not to plot the payoff

Details

Long Position: payoff = St - forward price

Short Position: payoff = forward price - StIf div.structure = "none"forward price= S ∗ er∗t

If div.structure = "discrete"

26 forward.prepaid

eff.i = er − 1

j = (1 + eff.i)1df − 1

Number of dividends: t∗ = t ∗ dfif k = NA: forward price = S ∗ er∗t−dividend∗s t∗|j

if k != j: forward price = S ∗ er∗t−dividend∗ 1−(1+k1+j )

j−k ∗ er∗t

if k = j: forward price = S ∗ er∗t−dividend∗ t∗

1+j ∗ er∗t

If div.structure = "continuous"forward price= S ∗ e(r−D)∗t

Payoff A data frame of different payoffs for given stock prices.

Price The forward price of the contract.

Leave an input variable as NA if it is not needed (ie. k=NA if div.structure="none").

See Also

forward.prepaid

Examples

forward(S=100,t=2,r=.03,position="short",div.structure="none")

forward(S=100,t=2,r=.03,position="long",div.structure="discrete",dividend=3,k=.02)

forward(S=100,t=1,r=.03,position="long",div.structure="continuous",D=.01)

forward.prepaid Prepaid Forward Contract

Description

Gives a table and graphical representation of the payoff of a prepaid forward contract, and calculatesthe prepaid forward price for the contract.

forward.prepaid(S,t,r,position,div.structure="none",dividend=NA,df=1,D=NA,k=NA,plot=FALSE)

forward.prepaid 27

Arguments

position either buyer or seller of the contract ("long" or "short")

div.structure the structure of the dividends for the underlying ("none", "continuous", or "dis-crete")

dividend amount of each dividend, or amount of first dividend if k is not NA

df dividend frequency- number of dividends per year

k dividend growth rate per df

plot tells whether or not to plot the payoff

Details

Long Position: payoff = St - prepaid forward price

Short Position: payoff = prepaid forward price - StIf div.structure = "none"forward price= S

If div.structure = "discrete"eff.i = er − 1

j = (1 + eff.i)1df − 1

Number of dividends: t∗ = t ∗ dfif k = NA: prepaid forward price = S−dividend∗a t∗|j

if k != j: prepaid forward price = S−dividend∗ 1−(1+k1+j )

if k = j: prepaid forward price = S−dividend∗ t∗

If div.structure = "continuous"prepaid forward price= S ∗ e−D∗t

Payoff A data frame of different payoffs for given stock prices.

Price The prepaid forward price of the contract.

Leave an input variable as NA if it is not needed (ie. k=NA if div.structure="none").

28 IRR

See Also

forward

Examples

forward.prepaid(S=100,t=2,r=.04,position="short",div.structure="none")

forward.prepaid(S=100,t=2,r=.03,position="long",div.structure="discrete",dividend=3,k=.02,df=2)

forward.prepaid(S=100,t=1,r=.05,position="long",div.structure="continuous",D=.06)

IRR Internal Rate of Return

Description

Calculates internal rate of return for a series of cash flows, and provides a time diagram of the cashflows.

IRR(cf0,cf,times,plot=FALSE)

Arguments

cf0 cash flow at period 0

times vector of the times for each cash flow

plot option whether or not to provide the time diagram

Details

cf0 =∑nk=1

cfk(1+irr)timesk

The internal rate of return.

Periods in t must be positive integers.

Uses polyroot function to solve equation given by series of cash flows, meaning that in the case ofhaving a negative IRR, multiple answers may be returned.

NPV 29

Author(s)

See Also

Examples

IRR(cf0=1,cf=c(1,2,1),times=c(1,3,4))

IRR(cf0=100,cf=c(1,1,30,40,50,1),times=c(1,1,3,4,5,6))

NPV Net Present Value

Description

Calculates the net present value for a series of cash flows, and provides a time diagram of the cashflows.

NPV(cf0,cf,times,i,plot=FALSE)

Arguments

cf0 cash flow at period 0

times vector of the times for each cash flow

i interest rate per period

plot tells whether or not to plot the time diagram of the cash flows

Details

NPV = cf0−∑nk=1

cfk(1+i)timesk

The NPV.

The periods in t must be positive integers.

The lengths of cf and t must be equal.

30 option.call

See Also

Examples

NPV(cf0=100,cf=c(50,40),times=c(3,5),i=.01)

NPV(cf0=100,cf=50,times=3,i=.05)

NPV(cf0=100,cf=c(50,60,10,20),times=c(1,5,9,9),i=.045)

option.call Call Option

Description

Gives a table and graphical representation of the payoff and profit of a long or short call option fora range of future stock prices.

option.call(S,K,r,t,sd,price=NA,position,plot=FALSE)

Arguments

K strike price

price specified call price if the Black Scholes pricing is not desired (leave as NA touse the Black Scholes pricing)

position either buyer or seller of option ("long" or "short")

Details

Long Position:

payoff = max(0, St −K)

profit = payoff - price∗er∗t

Short Position:

payoff = -max(0, St −K)

profit = payoff + price∗er∗t

option.put 31

Payoff A data frame of different payoffs and profits for given stock prices.Premium The price for the call option.

Finds the call price by using the Black Scholes equation by default.

Author(s)

See Also

option.put

bls.order1

Examples

option.call(S=100,K=110,r=.03,t=1.5,sd=.2,price=NA,position="short")

option.call(S=100,K=100,r=.03,t=1,sd=.2,price=10,position="long")

option.put Put Option

Description

Gives a table and graphical representation of the payoff and profit of a long or short put option fora range of future stock prices.

option.put(S,K,r,t,sd,price=NA,position,plot=FALSE)

Arguments

S spot price at time 0K strike pricer continuously compounded yearly risk free ratet time of expiration (in years)sd standard deviation of the stock (volatility)price specified put price if the Black Scholes pricing is not desired (leave as NA to

use the Black Scholes pricing)position either buyer or seller of option ("long" or "short")plot tells whether or not to plot the payoff and profit

32 perpetuity.arith

Details

Long Position:

payoff = max(0,K − St)

profit = payoff−price ∗ er∗t

Short Position:

payoff = -max(0,K − St)

profit = payoff+price ∗ er∗t

Premium The price of the put option.

Author(s)

See Also

option.call

bls.order1

Examples

option.put(S=100,K=110,r=.03,t=1,sd=.2,price=NA,position="short")

option.put(S=100,K=110,r=.03,t=1,sd=.2,price=NA,position="long")

perpetuity.arith Arithmetic Perpetuity

Description

Solves for the present value, amount of the first payment, the payment increment amount per period,or the interest rate for an arithmetically growing perpetuity.

perpetuity.arith(pv=NA,p=NA,q=NA,i=NA,ic=1,pf=1,imm=TRUE)

perpetuity.arith 33

Arguments

q payment increment amount per period

Details

ic − 1

j = (1 + eff.i)1pf − 1

Perpetuity Immediate:

pv = pj +

Perpetuity Due:

pv = (pj +qj2 ) ∗ (1 + j)

Returns a matrix of input variables, and calculated unknown variables.

One of pv, p, q, or i must be NA (unknown).

Author(s)

See Also

perpetuity.geo

perpetuity.level

annuity.arith

annuity.geo

annuity.level

Examples

perpetuity.arith(100,p=1,q=.5,i=NA,ic=1,pf=1,imm=TRUE)

perpetuity.arith(pv=NA,p=1,q=.5,i=.07,ic=1,pf=1,imm=TRUE)

perpetuity.arith(pv=100,p=NA,q=1,i=.05,ic=.5,pf=1,imm=FALSE)

34 perpetuity.geo

perpetuity.geo Geometric Perpetuity

Description

Solves for the present value, amount of the first payment, the payment growth rate, or the interestrate for a geometrically growing perpetuity.

perpetuity.geo(pv=NA,p=NA,k=NA,i=NA,ic=1,pf=1,imm=TRUE)

Arguments

pv present value

k payment growth rate per period

pf the payment frequency- number of payments and periods per year

imm option for perpetuity immediate or due, default is immediate (TRUE)

Details

ic − 1

j = (1 + eff.i)1pf − 1

j > k: pv = pj−k

Perpetuity Due:

j > k: pv = pj−k ∗ (1 + j)

One of pv, p, k, or i must be NA (unknown).

perpetuity.level 35

See Also

perpetuity.arith

perpetuity.level

annuity.arith

annuity.geo

annuity.level

Examples

perpetuity.geo(pv=NA,p=5,k=.03,i=.04,ic=1,pf=1,imm=TRUE)

perpetuity.geo(pv=1000,p=5,k=NA,i=.04,ic=1,pf=1,imm=FALSE)

perpetuity.level Level Perpetuity

Description

Solves for the present value, interest rate, or the amount of the payments for a level perpetuity.

perpetuity.level(pv=NA,pmt=NA,i=NA,ic=1,pf=1,imm=TRUE)

Arguments

pv present value

imm option for perpetuity immediate or annuity due, default is immediate (TRUE)

Details

ic − 1

j = (1 + eff.i)1pf − 1

pv = pmt ∗ a ∞|j = pmtj

Perpetuity Due:

pv = pmt ∗ a ∞|j = pmtj ∗ (1 + i)

36 protective.put

One of pv, pmt, or i must be NA (unknown).

Author(s)

See Also

perpetuity.arith

perpetuity.geo

annuity.arith

annuity.geo

annuity.level

Examples

perpetuity.level(pv=100,pmt=NA,i=.05,ic=1,pf=2,imm=TRUE)

perpetuity.level(pv=100,pmt=NA,i=.05,ic=1,pf=2,imm=FALSE)

protective.put Protective Put

Description

Gives a table and graphical representation of the payoff and profit of a protective put strategy for arange of future stock prices.

protective.put(S,K,r,t,sd,price=NA,plot=FALSE)

Arguments

S spot price at time 0K strike pricer continuously compounded yearly risk free ratet time of expiration (in years)sd standard deviation of the stock (volatility)price specified put price if the Black Scholes pricing is not desired (leave as NA to

use the Black Scholes pricing)plot tells whether or not to plot the payoff and profit

rate.conv 37

Details

For St <= K: payoff = K − S

For St > K: payoff = St − S

profit = payoff - price∗er∗t

Payoff A data frame of different payoffs and profits for given stock prices.Premium The price of the put option.

Author(s)

See Also

option.put

Examples

protective.put(S=100,K=100,r=.03,t=1,sd=.2)

protective.put(S=100,K=90,r=.01,t=.5,sd=.1)

rate.conv Interest, Discount, and Force of Interest Converter

Description

Converts given rate to desired nominal interest, discount, and force of interest rates.

rate.conv(rate, conv=1, type="interest", nom=1)

Arguments

rate current rateconv how many times per year the current rate is convertibletype current rate as one of "interest", "discount" or "force"nom desired number of times the calculated rates will be convertible

38 straddle

Details

1 + i = (1 + i(n)

n )n = (1− d)−1 = (1− d(m)

m )−m = eδ

A matrix of the interest, discount, and force of interest conversions for effective, given and desiredconversion rates.

The row named ’eff’ is used for the effective rates, and the nominal rates are in a row named’nom(x)’ where the rate is convertible x times per year.

Author(s)

Examples

rate.conv(rate=.05,conv=2,nom=1)

rate.conv(rate=.05,conv=2,nom=4,type="discount")

rate.conv(rate=.05,conv=2,nom=4,type="force")

straddle Straddle Spread

Description

Gives a table and graphical representation of the payoff and profit of a long or short straddle for arange of future stock prices.

straddle(S,K,r,t,price1,price2,position,plot=FALSE)

Arguments

K strike price of the call and put

price1 price of the long call with strike price K

price2 price of the long put with strike price K

straddle.bls 39

Details

Long Position:

For St <= K: payoff = K − St

For St > K: payoff = St −K

profit = payoff - (price1 + price2)∗er∗t

Short Position:

For St <= K: payoff = St −K

For St > K: payoff = K − St

profit = payoff + (price1 + price2)∗er∗t

Premiums A matrix of the premiums for the call and put options, and the net cost.

See Also

straddle.bls

option.put

option.call

strangle

Examples

straddle(S=100,K=110,r=.03,t=1,price1=15,price2=10,position="short")

straddle.bls Straddle Spread - Black Scholes

Description

Gives a table and graphical representation of the payoff and profit of a long or short straddle for arange of future stock prices. Uses the Black Scholes equation for the call and put prices.

straddle.bls(S,K,r,t,sd,position,plot=FALSE)

40 straddle.bls

Arguments

K strike price of the call and put

Details

Long Position:

For St <= K: payoff = K − St

For St > K: payoff = St −K

profit = payoff−(pricecall + priceput) ∗ er∗t

Short Position:

For St <= K: payoff = St −K

For St > K: payoff = K − St

profit = payoff+(pricecall + priceput) ∗ er∗t

See Also

option.put

option.call

strangle.bls

Examples

straddle.bls(S=100,K=110,r=.03,t=1,sd=.2,position="short")

straddle.bls(S=100,K=110,r=.03,t=1,sd=.2,position="long",plot=TRUE)

strangle 41

strangle Strangle Spread

Description

Gives a table and graphical representation of the payoff and profit of a long strangle spread for arange of future stock prices.

strangle(S,K1,K2,r,t,price1,price2,plot=FALSE)

Arguments

price1 price of the long put with strike price K1

Details

For St >= K2: payoff = St −K2

profit = payoff - (price1 + price2)∗er∗t

K1 < S < K2 must be true.

Author(s)

42 strangle.bls

See Also

strangle.bls

option.put

option.call

straddle

Examples

strangle(S=105,K1=100,K2=110,r=.03,t=1,price1=10,price2=15,plot=TRUE)

strangle.bls Strangle Spread - Black Scholes

Description

Gives a table and graphical representation of the payoff and profit of a long strangle spread for arange of future stock prices. Uses the Black Scholes equation for the call prices.

strangle.bls(S,K1,K2,r,t,sd,plot=FALSE)

Arguments

Details

For St >= K2: payoff = St −K2

profit = payoff−(priceK1 + priceK2) ∗ er∗t

swap.commodity 43

K1 < S < K2 must be true.

Author(s)

See Also

option.put

option.call

straddle.bls

Examples

strangle.bls(S=105,K1=100,K2=110,r=.03,t=1,sd=.2)

strangle.bls(S=115,K1=50,K2=130,r=.03,t=1,sd=.2)

swap.commodity Commodity Swap

Description

Solves for the fixed swap price, given the variable prices and interest rates (either as spot rates orzero coupon bond prices).

swap.commodity(prices, rates, type="spot_rate")

Arguments

prices vector of variable prices

rates vector of variable rates

type rates defined as either "spot_rate" or "zcb_price"

44 swap.rate

Details

For spot rates:∑nk=1

pricesk(1+ratesk)k

=∑nk=1

X(1+ratesk)k

For zero coupon bond prices:∑nk=1 pricesk ∗ ratesk =

∑nk=1 X ∗ ratesk

Where X = fixed swap price.

The fixed swap price.

Length of the price vector and rate vector must be of the same length.

Author(s)

See Also

swap.rate

Examples

swap.commodity(prices=c(103,106,108), rates=c(.04,.05,.06))

swap.commodity(prices=c(103,106,108), rates=c(.9615,.907,.8396),type="zcb_price")

swap.commodity(prices=c(105,105,105), rates=c(.85,.89,.80),type="zcb_price")

swap.rate Interest Rate Swap

Description

Solves for the fixed interest rate given the variable interest rates (either as spot rates or zero couponbond prices).

swap.rate(rates, type="spot_rate")

Arguments

rates vector of variable rates

type rates as either "spot_rate" or "zcb_price"

TVM 45

Details

For spot rates: 1 =∑nk=1[

R(1+ratesk)k

] + 1(1+ratesn)n

For zero coupon bond prices: 1 =∑nk=1(R ∗ ratesk) + ratesn

Where R = fixed swap rate.

The fixed interest rate swap.

See Also

swap.commodity

Examples

swap.rate(rates=c(.04, .05, .06), type = "spot_rate")

swap.rate(rates=c(.93,.95,.98,.90), type = "zcb_price")

TVM Time Value of Money

Description

Solves for the present value, future value, time, or the interest rate for the accumulation of moneyearning compound interest. It can also plot the time value for each period.

TVM(pv=NA,fv=NA,n=NA,i=NA,ic=1,plot=FALSE)

Arguments

pv present value

fv future value

n number of periods

i nominal interest rate convertible ic times per period

ic interest conversion frequency per period

plot tells whether or not to produce a plot of the time value at each period

Details

j = (1 + iic )

ic − 1

fv = pv ∗ (1 + j)n

46 yield.dollar

Exactly one of pv, fv, n, or i must be NA (unknown).

See Also

cf.analysis

Examples

TVM(pv=10,fv=20,i=.05,ic=2,plot=TRUE)

TVM(pv=50,n=5,i=.04,plot=TRUE)

yield.dollar Dollar Weighted Yield

Description

Calculates the dollar weighted yield.

yield.dollar(cf, times, start, end, endtime)

Arguments

times vector of times for when cash flows occur

start beginning balance

end ending balance

endtime end time of comparison

Details

I = end− start−∑nk=1 cfk

idw = I

start∗endtime−∑n

k=1cfk∗(endtime−timesk)

The dollar weighted yield.

yield.time 47

Time of comparison (endtime) must be larger than any number in vector of cash flow times.

Length of cashflow vector and times vector must be equal.

See Also

yield.time

Examples

yield.dollar(cf=c(20,10,50),times=c(.25,.5,.75),start=100,end=175,endtime=1)

yield.dollar(cf=c(500,-1000),times=c(3/12,18/12),start=25200,end=25900,endtime=21/12)

yield.time Time Weighted Yield

Description

Calculates the time weighted yield.

yield.time(cf,bal)

Arguments

bal vector of balances

Details

itw =∏nk=1(

bal1+k

balk+cfk)− 1

The time weighted yield.

Length of cash flows must be one less than the length of balances.

If lengths are equal, it will not use final cash flow.

Author(s)

48 yield.time

See Also

yield.dollar

Examples

yield.time(cf=c(0,200,100,50),bal=c(1000,800,1150,1550,1700))

∗Topic amortizationamort.period, 2amort.table, 4

∗Topic analysisbond, 13cf.analysis, 19

∗Topic annuityannuity.arith, 5annuity.geo, 7annuity.level, 8

∗Topic arithmeticannuity.arith, 5perpetuity.arith, 32

∗Topic bondbond, 13

∗Topic callbear.call, 9bear.call.bls, 11bls.order1, 12bull.call, 15bull.call.bls, 16butterfly.spread, 17butterfly.spread.bls, 18collar, 20collar.bls, 22covered.call, 23option.call, 30straddle, 38straddle.bls, 39strangle, 41strangle.bls, 42

∗Topic forwardforward, 25forward.prepaid, 26

∗Topic geometricannuity.geo, 7perpetuity.geo, 34

∗Topic greeksbls.order1, 12

∗Topic interestrate.conv, 37swap.rate, 44

∗Topic irrIRR, 28

∗Topic levelannuity.level, 8perpetuity.level, 35

∗Topic optionbear.call, 9bear.call.bls, 11bls.order1, 12bull.call, 15bull.call.bls, 16butterfly.spread, 17butterfly.spread.bls, 18collar, 20collar.bls, 22covered.call, 23covered.put, 24option.call, 30option.put, 31protective.put, 36straddle, 38straddle.bls, 39strangle, 41strangle.bls, 42

∗Topic perpetuityperpetuity.arith, 32perpetuity.geo, 34perpetuity.level, 35

∗Topic putbls.order1, 12collar, 20collar.bls, 22covered.put, 24option.put, 31protective.put, 36straddle, 38

50 INDEX

straddle.bls, 39strangle, 41strangle.bls, 42

∗Topic spreadbear.call, 9bear.call.bls, 11bull.call, 15bull.call.bls, 16butterfly.spread, 17butterfly.spread.bls, 18collar, 20collar.bls, 22straddle, 38straddle.bls, 39strangle, 41strangle.bls, 42

∗Topic swapswap.commodity, 43swap.rate, 44

∗Topic timeTVM, 45yield.time, 47

∗Topic valuecf.analysis, 19NPV, 29TVM, 45

∗Topic yieldIRR, 28yield.dollar, 46yield.time, 47

amort.period, 2, 5amort.table, 3, 4annuity.arith, 5, 8, 9, 33, 35, 36annuity.geo, 6, 7, 9, 33, 35, 36annuity.level, 5, 6, 8, 8, 33, 35, 36

bear.call, 9, 12, 16, 17bear.call.bls, 10, 11bls.order1, 12, 31, 32bond, 13bull.call, 10, 15bull.call.bls, 12, 16, 16butterfly.spread, 17, 19butterfly.spread.bls, 18, 18

cf.analysis, 19, 46collar, 20collar.bls, 21, 22

covered.call, 23, 25covered.put, 23, 24

forward, 25, 28forward.prepaid, 26, 26

IRR, 28, 30

NPV, 29, 29

option.call, 10, 12, 13, 16–19, 21–23, 30,32, 39, 40, 42, 43

option.put, 13, 21, 22, 25, 31, 31, 37, 39, 40,42, 43

perpetuity.arith, 6, 8, 9, 32, 35, 36perpetuity.geo, 6, 8, 9, 33, 34, 36perpetuity.level, 6, 8, 9, 33, 35, 35protective.put, 36

rate.conv, 37

straddle, 38, 42straddle.bls, 39, 39, 43strangle, 39, 41strangle.bls, 40, 42, 42swap.commodity, 43, 45swap.rate, 44, 44

TVM, 20, 45

yield.dollar, 46, 48yield.time, 47, 47

package ‘financialmath’ - r · december 16, 2016 type package title financial mathematics for...

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